The value of λ such that sum of the squares of the roots of the...

We have, ${x}^{2} + ( 3 - \lambda ) x + 2 = \lambda, \text{have roots}\alpha \text{ and } \beta .$

Then, $f ( \lambda ) = {\alpha}^{2} + {\beta}^{2}$

$= {\left(\alpha + \beta\right)}^{2} - 2\alpha\beta = {\left(3 - \lambda\right)}^{2} - 2\left(2 - \lambda\right) = {\lambda}^{2} - 6\lambda + 9 - 4 + 2\lambda = {\lambda}^{2} - 4\lambda + 5$

${f}^{′}\left(\lambda\right) = 2\lambda - 4 = 0$ ${\text{ for }\text{ the }\text{ minimum }\text{ value }\text{ of } f\left(\lambda\right) }$

As ${f}^{''} ( \lambda ) = 2 > 0$

$⇒ \lambda = 2$